I'm not familiar with this reference, but what it sounds like to me is that the wound was severe, and eventually he was expected to die from it.
The dying process from deep wounds was slow back in feudal times. Their medicine was primitive. They didn't have modern surgical techniques or antibiotics. They would invariably develop a serious infection, which often took their lives. They would go into shock from sepsis. Their appendages would gangrene and would have to be chopped off to prevent death (and that might be too embarrassing for a samurai who relies on his arm for wielding a sword). Their stomach and intestines, if punctured, might leak causing severe pain and an agonizing death.
The man would be in agony, crying, and delirious. He would lose his mind. It would be a long and emotional sight. For samurai who value self-control and keeping a calm mind, this alone would be embarrassing enough to ask for death instead.
What his father wanted was for his son to die "honorably" rather than die like this from a wound caused by his enemy. It would also mean that his duel was not a tie, but instead he won the duel, if he didn't die from it.
Now that is nonsense thinking, of course. Technically, if you committed suicide because of a mortal wound, then the real cause of your death is still the wound. Everyone knows it. Even the samurai. They weren't stupid.
So the only thing left to conclude is that dying quickly from dismemberment was a preferable way to die compared with a long, lingering, agonizing death. It would save everyone around him from the emotional pain as well.
The tie-in with samurai code and spiritualism is just the icing on the cake. It's the religion to make the medicine more palatable. It transmutes a really bad way to die into a good way, one where he is showing how strong he is mentally and spiritually.
A lot of things about the samurai way of life is incomprehensible to us westerners in modern times. Suffice it to say, there's a reason for everything. And as a side note, just because there's a reason, it doesn't mean it's a rational one. It's okay to be critical of them.
My thoughts anyway.